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I need some light in this exercise:

"If $f:\mathbb {R^n}\rightarrow \mathbb {R^n}$ is continuous differentiable and the image of $f(\mathbb {R^n})$ has $\text{int} =\varnothing$ then the determinant of the Jacobian matrix is zero."

I really don't know where to start... I appreciate any help that helps me to get started.

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  • $\begingroup$ I haven't done this in ages, but it seems obviously true: I suggest check definition of the jacobian as a limit. With the interior of the image empty, the only points converging to a a point $x_0$ will be $x_0$ themselves, so the derviative will be zero. $\endgroup$ – gnometorule Dec 11 '12 at 16:26
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If the determinant of the Jacobian was non-zero at some point, then the inverse function theorem would show that it is locally invertible, and hence maps a small open set into an open set, which would contradict the image having empty interior.

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Suppose there is a point $x$ such that $\operatorname{det}(f'(x))\neq 0$. What happens if you use the inverse function theorem?

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